The principle of multiplication is a fundamental concept in combinatorics and probability theory. It states that if there are \(n_1\) ways to do one thing and \(n_2\) ways to do another, then there are \(n_1 \times n_2\) ways to do both.
Mathematically, if an event E1 can occur in \(n_1\) ways, and for each of these ways, an event E2 can occur in \(n_2\) ways, then the total number of ways both events E1 and E2 can occur together is \(n_1 \times n_2\).
For example, let's consider two events:
- Event E1: Rolling a fair six-sided die and getting an even number.
- Event E2: Tossing a fair coin and getting heads.
Event E1 can occur in three ways (rolling a 2, 4, or 6), and event E2 can occur in two ways (getting heads or tails). According to the principle of multiplication, the total number of ways both events can occur together is \(3 \times 2 = 6\). These ways are: (2, H), (2, T), (4, H), (4, T), (6, H), (6, T).
This principle is widely used in counting problems where the occurrences of different events are independent of each other. The total number of outcomes for a sequence of independent events is the product of the number of outcomes for each individual event.
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